Asymptotic behavior of Aldous' gossip process

Aldous [(2007) Preprint] defined a gossip process in which space is a discrete $N\times N$ torus, and the state of the process at time $t$ is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate $N^{-α}$ to a site chosen at random from the torus. We will be interested in the case in which $α<3$, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically $T=(2-2α/3)N^{α/3}\log N$. If $ρ_s$ is the fraction of the population who know the information at time $s$ and $\varepsilon$ is small then, for large $N$, the time until $ρ_s$ reaches $\varepsilon$ is $T(\varepsilon)\approx T+N^{α/3}\log (3\varepsilon /M)$, where $M$ is a random variable determined by the early spread of the information. The value of $ρ_s$ at time $s=T(1/3)+tN^{α/3}$ is almost a deterministic function $h(t)$ which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.

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